Electromagnetic Imaging and Simulated Annealing

In contrast with acoustical imaging methods, for which the wave field is dominated by propagation effects, electromagnetic imaging of conductive media suffers from the diffusive behavior of the electromagnetic field. An important question to address when working toward the achievement of electromagnetic imaging concerns the possibility of resolving the diffusion damping. Exact inversion will be looking at the solvability of the integral equation relating a diffusive field to its dual wavefield. This equation is ill posed because its Laplace-like kernel makes the inverse problem of finding the dual wave field a notoriously difficult (both numerically and mathematically) one. Stochastic inversion is another alternative based on least squares fitting. In this inverse problem approach, extracting the wave field is still a relatively instable process, although the L2 misfit function for data without noise presents a global minimum. The simulated annealing overcomes this instability for parameterization of this problem designed as follows. The unknown wave field is expected to be a sequence of impulsive functions. The number of impulsive functions can be determined by using a statistical criterion, called AIC, which comes from the Prony technique. The simulated annealing is applied to the positions of the reflections, while the amplitudes, which are not taken as parameters, are obtained by linear fitting. The simulated annealing method proves to be efficient even in the presence of noise. Furthermore, this nonlinear numerical inversion furnishes statistical quantities which allows an estimation of the resolution. Simple synthetic examples illustrate the performance of the inversion, while a synthetic finite element example shows the final pseudo-seismic section to be processed by standard seismic migration techniques.